3.30 \(\int \cosh (a+\frac {b}{x^2}) \, dx\)

Optimal. Leaf size=67 \[ \frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \cosh \left (a+\frac {b}{x^2}\right ) \]

[Out]

x*cosh(a+b/x^2)+1/2*erf(b^(1/2)/x)*b^(1/2)*Pi^(1/2)/exp(a)-1/2*exp(a)*erfi(b^(1/2)/x)*b^(1/2)*Pi^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5303, 5327, 5298, 2204, 2205} \[ \frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )+x \cosh \left (a+\frac {b}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b/x^2],x]

[Out]

x*Cosh[a + b/x^2] + (Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/(2*E^a) - (Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/2

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5298

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5303

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.), x_Symbol] :> -Subst[Int[(a + b*Cosh[c + d/x^n])^p/x^2
, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]

Rule 5327

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[((e*x)^(m + 1)*Cosh[c + d*x^n])/(e*(m +
1)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[
n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=x \cosh \left (a+\frac {b}{x^2}\right )-(2 b) \operatorname {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=x \cosh \left (a+\frac {b}{x^2}\right )+b \operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-b \operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 71, normalized size = 1.06 \[ \frac {1}{2} \sqrt {\pi } \sqrt {b} (\cosh (a)-\sinh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} (\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \cosh \left (a+\frac {b}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b/x^2],x]

[Out]

x*Cosh[a + b/x^2] + (Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]))/2 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[b]/x
]*(Cosh[a] + Sinh[a]))/2

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fricas [B]  time = 0.69, size = 225, normalized size = 3.36 \[ \frac {x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + \sqrt {\pi } {\left (\cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } {\left (\cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) + 2 \, x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + x}{2 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x^2),x, algorithm="fricas")

[Out]

1/2*(x*cosh((a*x^2 + b)/x^2)^2 + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) + cosh((a*x^2 + b)/x^2)*sinh(a) + (co
sh(a) + sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(-b)*erf(sqrt(-b)/x) + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) - c
osh((a*x^2 + b)/x^2)*sinh(a) + (cosh(a) - sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(b)*erf(sqrt(b)/x) + 2*x*cosh((a
*x^2 + b)/x^2)*sinh((a*x^2 + b)/x^2) + x*sinh((a*x^2 + b)/x^2)^2 + x)/(cosh((a*x^2 + b)/x^2) + sinh((a*x^2 + b
)/x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \left (a + \frac {b}{x^{2}}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x^2),x, algorithm="giac")

[Out]

integrate(cosh(a + b/x^2), x)

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maple [A]  time = 0.13, size = 70, normalized size = 1.04 \[ \frac {\sqrt {b}\, {\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {b}}{x}\right )}{2}+\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}} x}{2}+\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}} x}{2}-\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{2 \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b/x^2),x)

[Out]

1/2*b^(1/2)*exp(-a)*Pi^(1/2)*erf(b^(1/2)/x)+1/2*exp(-a)*exp(-b/x^2)*x+1/2*exp(a)*exp(b/x^2)*x-1/2*exp(a)*b*Pi^
(1/2)/(-b)^(1/2)*erf((-b)^(1/2)/x)

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maxima [A]  time = 0.37, size = 72, normalized size = 1.07 \[ \frac {1}{2} \, b {\left (\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {b}{x^{2}}}\right ) - 1\right )} e^{\left (-a\right )}}{x \sqrt {\frac {b}{x^{2}}}} - \frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {b}{x^{2}}}\right ) - 1\right )} e^{a}}{x \sqrt {-\frac {b}{x^{2}}}}\right )} + x \cosh \left (a + \frac {b}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x^2),x, algorithm="maxima")

[Out]

1/2*b*(sqrt(pi)*(erf(sqrt(b/x^2)) - 1)*e^(-a)/(x*sqrt(b/x^2)) - sqrt(pi)*(erf(sqrt(-b/x^2)) - 1)*e^a/(x*sqrt(-
b/x^2))) + x*cosh(a + b/x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cosh}\left (a+\frac {b}{x^2}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b/x^2),x)

[Out]

int(cosh(a + b/x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x**2),x)

[Out]

Integral(cosh(a + b/x**2), x)

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